Answer :
To determine how much Aaron should pay in the first month, let's break down the problem step-by-step:
1. Define the total amount borrowed: Aaron borrows [tex]$150 from his friend. 2. Represent the payments Aaron needs to make: - Let \( x \) be the amount Aaron pays in the first month. - According to the problem, he pays half the amount he paid in the previous month for each subsequent month. Therefore: - First month's payment: \( x \) - Second month's payment: \( \frac{x}{2} \) - Third month's payment: \( \frac{x}{4} \) - Fourth month's payment: \( \frac{x}{8} \) 3. Set up an equation for the total amount: Aaron promises to repay the total amount of $[/tex]150 in these 4 installments. So, we sum up the payments and set them equal to [tex]$150: \[ x + \frac{x}{2} + \frac{x}{4} + \frac{x}{8} = 150 \] 4. Sum the series: This expression is a geometric series where each term is half of the previous one. The series \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} \) can be summed up as follows: \[ 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{15}{8} \] 5. Express the total payment in terms of \( x \): \[ x \left( \frac{15}{8} \right) = 150 \] 6. Solve for \( x \): \[ x = 150 \times \frac{8}{15} \] \[ x = 80 \] Therefore, Aaron should pay \$[/tex]80 in the first month to repay the money as scheduled.
Thus, the correct answer is:
C. \$80
1. Define the total amount borrowed: Aaron borrows [tex]$150 from his friend. 2. Represent the payments Aaron needs to make: - Let \( x \) be the amount Aaron pays in the first month. - According to the problem, he pays half the amount he paid in the previous month for each subsequent month. Therefore: - First month's payment: \( x \) - Second month's payment: \( \frac{x}{2} \) - Third month's payment: \( \frac{x}{4} \) - Fourth month's payment: \( \frac{x}{8} \) 3. Set up an equation for the total amount: Aaron promises to repay the total amount of $[/tex]150 in these 4 installments. So, we sum up the payments and set them equal to [tex]$150: \[ x + \frac{x}{2} + \frac{x}{4} + \frac{x}{8} = 150 \] 4. Sum the series: This expression is a geometric series where each term is half of the previous one. The series \( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} \) can be summed up as follows: \[ 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{15}{8} \] 5. Express the total payment in terms of \( x \): \[ x \left( \frac{15}{8} \right) = 150 \] 6. Solve for \( x \): \[ x = 150 \times \frac{8}{15} \] \[ x = 80 \] Therefore, Aaron should pay \$[/tex]80 in the first month to repay the money as scheduled.
Thus, the correct answer is:
C. \$80
